The Dirichlet Boundary Conditions and Related Natural Boundary Conditions in Strengthened Sobolev

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The Dirichlet Boundary Conditions and Related Natural Boundary Conditions in Strengthened Sobolev

Spaces ^for Discretized Parabolic Problems

E. D’YAKONOV

Departmentof^ComputerMathematicsandCybernetics, Moscow StateUniversity,Moscow, 119899,Russia (Received21April1999)

Correctnessofinitialboundaryvalueproblemsand their discretizations areanalyzedunder unusual second-orderboundary conditions,which canbe considered asnatural boundarycon- ditions instrengthened Sobolev spacesand asimprovements(insomecases)ofthe classical Dirichletboundaryconditions.Specialattention ispaid to optimal perturbationestimatesfor new variantsofthepenaltymethod withrespecttothe Dirichlet conditions.

Keywords." TheDirichletand related naturalboundaryconditions,Strengthened Sobolevspaces, Parabolicproblems

0. INTRODUCTION

The Dirichlet boundary conditions are of funda- mental importance in mathematical physics and other fields of science. They serve as a means to isolate the problem underconsideration in agiven domain f,

c

R^d from the outside world; in what follows, we assume that ^$2 is a bounded domain with Lipschitz piecewise smooth boundaryP.

Other boundary conditions arepossible. Some- timestheyleadto abetter description of what hap- pensonthe boundary;inthe case, forexample, of anellipticvariationalprobleminthe Sobolev space

H ^(f) W ^(f*),

^there is a common opinion that the homogeneous Dirichlet conditions can be treated in terms of the penalty method as a limit

269

of natural boundary conditions of the type

(Ou/Off) + ⁽¹ + ^1/c)u

0, where ff is the unit vector of the outer normal to the boundary, c /0

(see

Courant, 1943; Babushka, 1973; 1987;

Sobolevskii,1981;Bramble, 1981;Glowinski,

1983).

These conditions areconnectedwiththeadditional term(penalty

term) F(u) (1 + 1/e)lul

²

(I,10,

²

-=

Ilull cc))

in the minimizedenergyfunctional.Some- times conditionsof suchatype haveagood physical senseandimprovetheoriginalDirichlet conditions this isthe case, forexample,intheoryofelasticity when springson the boundaryare allowed; some- times the new conditions are considered as very artificial(see,forexample,Glowinski

(1983),

where theywere appliedtoproblems of hydrodynamics).

From themathematicalpoint of view, thepenalty

term might be also considered as rather weak because of additional smoothnessrequirements on thesolutionsforobtainingestimatesof the type

Itwasshown recently

(see

D’yakonov, 1997a,b) that suchandevenstrongerestimates canbeproved assuming onlycorrectness of the original problem ifapply thepenalty

term/>(u) (1 + ^l/e)

2

with u

>_ 1/2

^and

treat

the arising problem in the corresponding strengthened Sobolev space. The case u- is the most remarkable since iteven al- lows some domains with slits and has good pers- pectives from the computational point of view;

moreover, it often has an obvious physical sense.

For example, in theory of elastic membranes, it corresponds to the presence ofstring(stiffener) ^on theboundary;similarproblemsforplates andshells with stiffeners are of special importance in many applications

(see

Courant, 1943;D’yakonov, 1996a;

Ciarlet, 1997 and references therein); ^it should be noted that S.P. Timoshenko was the first to set them(in^apre-Hilbertspace)in1915;certainprob- lems ofhydrodynamicswithsurfacetensioncanbe also set in similarstrengthenedSobolevspaces

(see

D’yakonov, 1997c);importanceof relevantsecond- order boundary conditions was also underlined in thestudy ofproblemsonunboundeddomains

(see

DuongandJoly, 1994; Sheen,

1993).

Thegoalof thepresent paperisageneralization ofthe approach indicated above to the case ofpa- rabolicproblems. Correctnessof the initialbound- ary value problems (with ^an emphasis on their discretizations withrespecttothetime)^{is analyzed} by the energy method under specialchoice ofenergy spaces associated with the strengthened Sobolev spaces.

Specialattention ispaidtooptimal perturbation estimates for the variant of the penalty method mentioned above (with respect to the the homo- geneousDirichletconditions). Theresults obtained also yield understanding of the mechanism of splitting of the parabolic problem into separate

ones in subdomains with the homogeneous Dirichlet conditions on the boundaries of these subdomains.

In this paper, only real Hilbert spaces and bounded operators are used; the normed linear space oflinearbounded operators mapping Uinto F is denoted by

(U;F); IILII

^L

IU>-+F

sup#0

IILVllFIJVllb; (H) (H; H);

^KerL=

{v:

^Lv-

0}

thekernel(null-space)of theoperator L; Im

L--L{ U}-

theimage(range) ofthe opera- tor L; /-the identity operator; H*-the linear space ofbounded linear functionals mapping H into

R;

A*-theadjoint operatortoA

(H; H2) ((AIg, P)H --(u,A*v)H ,,

^’quEH,,

vEH2); A.-the

symmetric partofA

(H),i.e.,A.,.- 2-(A +

A*);

.+(H)

denotes the set of linear, symmetric, and positive definite operators in

(H); H(B)--the

Hilbertspacediffering fromHonly byinnerprod- uct defined by B

+(H),

^namely (u,

V)H(B) (u,

V)B--(Bu, V)H--(Bu,

v).

Fornonnegativefunctions

f(h)

and g(h),

f(h)

g(h) implies that there exist positive constants ₀ and _l such that og(h)<f(h)<g(h); we also make use of this notation for norms, quadratic functionals and operators, e.g., L IL

+ ^(H).

^For simplicity, ^we consider only 2

c R2;

the spacevariables are denotedby

x

^and x2with X__[X X2];

[U]2

_0,

]]u

²_L2(f)’"the time variable is 1--X₀ [0, T];

. ^[X0,

^X1,

^X2]

^{e Qr f x}

^{[0, T];}

D,u Ou/Ox,s,,s 0, 1,2;

[Vu]

[(Dlb/)²

-- (D2tA)2]I/2;

(H,

V)I,"

(DlU,

DlV)o,a+(D2u, D2v)o,a; lull,fz Vul

²

1)0,s

1/2

"

0.1. Variational Problems with Linear Constraintsand Corresponding

Problems with Strongly Saddle Operators We briefly recall the most important facts related to variational problems with linear constraints (see,Brezzi, 1974;D’yakonov, 1996aandreferences therein)^inagivenHilbertspace

H.

The variational

problemis tofind

ul argmin

[/2(vl) 21(vl)]

EV

with aquadratic

functional/2(vl) Ilv1112/41

^{and lin-}

ear functional E

Hi;

^{the space}

V1

of admissible functions consists of functions_vlsuchthat

L2,1 v

0foragiven

L2,1 (H1;H2)

thatis

V1

^Ker

L2,1.

Thestandard penalty method for

(0.1)

consists in settingasequenceof unconstrainedproblems

ul,c arg_vlmin_EH1

[/2(v,) +

l/gilL2

1/111/2 ^21(Vl)]

(0.2)

withThe classicalthepenalty parameterLagrange approache

--

^/0.

(the Lagrange

multiplier

method)

^{to the}problem

(0.1)

^{is to}replace itbytheproblem

Lu

L2,1

⁰ u2

a Hilbert space

H-H1

^xH2, where the addi- tional function_u2plays the roleoftheLagrangian multiplier,

L*

(0.4)

Ll,1

^I1,

Ll,2

2,1"

The modern formulation ofproblems

(0.3), (0.4)

(saddle-point problems arising from Lagrangian multipliers)and moregeneralonesof type

[ ^{LI,1 L1,2}

Lcuc _{L2,1 -eI2}

(see (0.2))

^is based on the use ofspecial types of operators

L2,1

^{which canbe} described asnormally invertible operators (operators withIm

L2,1--H2);

equationswiththemarecalled everywhere solvable

(see

Krein,

1971).

They correspond to aparticular and very well known case of normally solvable operators

(see

Krein, 1971; Rempel and Schulze, 1985; Trenogin, 1980;D’yakonov, 1996aandrefer- ences therein)which are defined as operatorswith

ImL2,1

^{being subspaces in}

H2

^{(operators with}

closed images); if

L2,1

^{is a} normally solvable operator, then

H

^{is anorthogonalsumofKer}

L2,1

andIm

L,,

^i.e.,

H1

^Ker

L2,1

^@Im

L1,2. (0.6)

Anormallyinvertibleoperator

L2,1

yieldsa one- to-one mapping of the Hilbert space

ImL,2

(orthogonal complementⁱⁿ

H1

^toKer

L2,)

^into

H2 (see (0.6))

and by the Banach theorem this map- ping is invertible and the corresponding inverse

L2,1(-1) L,I

^{is such that}

]lL,l ^{0--1 <}

^c.

^(0.7)

Wenotethat thewell-known inf-sup condition,

(L2,1Ul,U2)H2 _>

0-

>

O, inf sup

/A2H2/./lHl

]l///1 IIHl ^II/A2 IH2

isoften usedinsteadof

(0.7); (0.8)

canbewritten in the form

82,

Vu H (0.9)

(seeD’yakonov, 1996a; Giraultand Raviart, 1986;

Pironneeau, 1989andreferencestherein)^{which im-} plies

(0.6).

^Itis alsoworth noting that

(0.7)

yields inclusions

spL2,1L,

^C

[0-2, ilL2,ll2],

sp

L, ^{L2, \0 < [0 -2, I{L2,} [12],

where spA denotes the spectrum of A (see D’yakonov, 1986); for

L2,

associated with the divergence operators, first results (in pre-Hilbert spaces) about such spectrums were obtained in Cosserat

(1898).

Theoperator

L

ⁱⁿ

^(0.5)

^{with e}

^>

^{0 and anormally}

invertible operator

L2,1

^is called a strongly saddle operator. For such operators, it was proved

(see

D’yakonov, 1983;

1996a)

that

L

is invertible and

IIL - II

^K,

^(0.10)

where theconstantKcan bechosenuniformly for alle

>

^{0. This}impliesthatproblem

(0.5) (or (0.3))

^is correctlyposedand the firstcomponentof itssolu- tion coincides with the solution ofproblem

(0.2) (or (0.1)).

Moreover

(0.10)

implies,for all param- eters indicated, that

foragivenc0

>

^0.

Thanks to anunderstanding of the role of

(0.8)

and itsgridanalogsin thetheory of progective-grid (finite element) methods and iterative processes, it now seems reasonable to regard problems

(0.5)

as basic and, instead ofproblems

(0.2)

and more general ones involving a large parameter l/c, ^to work with correspondingproblems

(0.5) (see,

e.g., D’yakonov,

1996a).

The resultsindicatedholdformoregeneral prob- lems of type

ILl,1 ^L1,2 [Ul,e

Leue L2,1 -cL2,2

u2,e

with

L2,2 _>

^0.

Moreover,

thecorrespondingcorrect- ness and perturbation theorems

(see (0.10)

and

(0.11))

can be obtainedforproblems of type

(0.12)

involving nonsymmetric operators

LI,1

and

L2,2

with theirsymmetric parts

Ll,l,s

^and

Lz,2,,

respec- tively.Itsuffices to

assume

that

L,I,

^/2+

(H1)

^and

L2,2,, >

⁰

(see

D’yakonov,

1996a).

0.2. StrengthenedSobolev Spacesand Perturbation of the Dirichlet Conditions The most significant feature of the problems we study here is that they involve setting in strength- ened Sobolev spaces

Gl,m(t2; F)

^=_

G,m ^(F

^Oft,m

>

1/2)insteadof theclassical Sobolevspace

W21 ^(ft)

H

(ft)(see

D’yakonov, 1996a; 1997a,b,c,d;Ciarlet, 1997and referencestherein). In^{the caseofsmooth} F wedefine

Gl,m

^{as a}subset of functions in

HI(Q)

suchthat their traces onFbelongto

W[(F),

^sowe

maydefinethenormby

ivll2

al,m

IIv I1,Ut

²

-- ^{IITrr vii}

^2w/(r)

^(0.13)

(similar spaces defined in terms of Fourier trans- formations of the corresponding extended func- tions were considered in Vishik, 1970). We emphasize that the trace operator Tr

Trr

^{is un-}

derstood in the standard way as an element of

/2(HI(Q);L2(F’)).

IfP is not smooth but consists of several smooth arcs

F;

then

(0.13)

should be replaced by

(0.14) Thisspace

Gl,m

^{is aHilbertspace and tracesofits} elementsonFcan beconsidered as continuous func- tions(almosteverywhere,seeD’yakonov, 1997c,d).

The most important ^case of

(0.13)

^and

(0.14)

is connectedwithm- when

Ilvll

²

GI,1- IIFII1,Q -+-

^ly

,F’ ^(0.15)

v

12,r ^I[Trr;vll 2Hl(Ii)," ^(0,16)

this Hilbert space of traces will be denoted by

G2([[’) G2

^{andweprefertowrite}

G1

^insteadof

Gl,.

The fundamental for the result of our analysis is connected with consideration of restriction of the operatorTr

e (H (f2); Lz(F))

^toourspace

G ^(we

denote it bythe same symbol); thisrestriction

(see

D’yakonov,

1997b)

issuch that

Tr

/2(G1; G2),

ImTr

G. (0.17)

Tospecify applications of theapproachindicated to the homogeneousDirichletconditions,we start by indicating that

H ^{(ft) W (ft)}

can be consid- eredas asubspace

V1

^ofourstrengthened Sobolev space

Gl,l(ft; F) G1 c Hl(ft) (see (0.15)-(0.17)).

Hence,

the originalvariationalproblem

//1 arg min

[I2 (F1) 2/(Vl)], (0.18)

with

I(v) [Ivll,f, ^{H(2) (0.19)}

can be easily reformulated as problem

(0.1)

in the space

V1 c G

^with

where

H Gl,(f;

Y),

H2- H(V), bl,l(Ul;

Vl) ^is symmetric,

bl,l(Vl’Y1) 2(Vl) ]v1112

_Hi

b2,2(H2; F2) ("2, P2)H2, ^(1.2)

/2(Vl) --I20(Vl)

^@

_IIVl

^{2 2}

(0.20)

/91,2(/32; Vl) b2,1(F1; H2) (Wr

^v,,

u2),,p,

VVl

^{E HI,}

Vu2 H2. (1.3)

or even

Vl

12

^1,Y

IlVl I[GI

²

^{(0 21)}

Now,onthebasisof

(1.1)-(1.3)

and as atypical example of nonstationaryproblems,weconsider a sequence of stationaryproblems

The second example is connnected with the origi- nalproblem(see

(0.18))

under a moregeneral than

(0.19)

condition oftype

(0.22)

Then

(0.20)

holds.

Finally,under

(0.22)

^{it is}possibletotakea sub- space

G(

^c

G

ⁱⁿ the role of

H

ⁱⁿ

^(0.1)

^{and deal}

with

(0.21).

This is the case ifelements of

G

^are

suchthat

(v, 1)0,r

⁰

(see

D’yakonov, 1997a,b).

1. DISCRETIZED IN TIME PARABOLIC PROBLEMS

1.1. OriginalBasic ParabolicProblems inStrengthened SobolevSpaces

For consideration of nonstationary problems, it is convenient to rewrite the related stationary problems

(0.2)

and

(0.5)

using bilinear forms

br,l,

defined and bounded on

Hz Hr

and connected with the operators

Lr,!

^{by the} standard equalities

br,(v; vr)=(L,v,v),

r[1,2], l[1,2]. Then

(0.5)

in case of(0.1),

(0.18)-(0.20)

^isjust theproblem of findingu EHsuch that

bl,l(Ul; Vl)

^q-

bl,2(u2; Vl) /l(Vl),

b2,1 (Ul; v2) cb2,2(u2; v2)

^0,

VV2

H2,

(1.1)

n+l refers to arbitrary wherer T/n,1+

H,

^v

elements of

H,.,

r 1,2,n 0,..., n* 1.

It is well known that if ^we take here

H

H0 ^(f) W ^(f)

and c-0, then

(1.4)

corresponds to thewell-known implicit semidiscretization with respect to ofa parabolic equation

Dou + Lu =f

withthehomogeneous Dirichlet conditions on the lateral surfaceofthe cylinderQT f x

[0, T]

^(spa-

tial variables remain continuous). Thus

(1.4)

^with

H- ^{G,(2; F)}

^{can betreated asa}perturbationof the nonstationary problemwith the homogeneous Dirichlet conditions.Weconcentrate onthese prob- lems to attain the desired similarity with the sta- tionaryonesand avoidintroduction ofnew spaces that arise in dealing with continuous and cor- respond to special strengthening of the Sobolev space

H(Q,).

1.2. A Priori Estimates and Correctness

Hereafter,

H

^and

H2

^{are Hilbert} spaces, and the original problem

(see (1.4)

^{with fixed}

n)

^is formu- lated in the Hilbert space

H-H

^x

H2

^{as the}

operator equation

m0[u

⁺¹

u]/7. + LI,lU

⁺¹

+ L1,2u

2

with

Vul ^c

^H1,

VVl ^c (1.6)

(f"+, vl)i4, ln+(vl), Vv

^{HI, and}

u

^{-0 (this}

can be assumed without loss of generality). Note that

Mo (H1;H1)

and M-M*> 0; moreover,

THEOREM 1.1 Let

HI-- G,l(f;P), Hz--HI(E),

and operatorLin

(0.3)

withc

>_

0beastronglysad- dle operatorin the space

H-H1 H2.

^Then

1/7.Mo + LI,1 L1,2

1

L2,1 -cI2 (1.7)

with

Mo from ^(1.6)

^{is also a strongly} saddle oper- atorin the spaceH.

Proof

^Since

Mo ,(H1; HI)

^{and M M*}

>

0, we conclude that

[1/7.M0 + LI,1] -I.

^Therefore,

L

^from

^(1.7)

^hasthe desiredproperties.

Theorem 1.1 impliesthatproblem

(1.5)

and even moregeneral problem

(1.8)

with

f2

^{H2,n-1,..., k}

^<_

^{n*-T/-,has aunique}

solution

(here

and elsewhere, we prefer to write

Oou

^insteadof

^{[uff u-l]/7..}

Note also that problem

(1.5)

^with e>0 is reducedto

M00u

⁺¹

^{+ LI,1} uff+l ^-+--Ll,zL2

^lU

+l ^/ln+l

(1.9)

this type of problem is often used in the study of the Stokes problems

(see

D’yakonov, 1996a;

Kobelkov, 1994; Sobolevskiiand Vasil’ev,

1978).

THEOREM 1.2 Let the conditions

of

Theorem 1.1 be

satisfied.

^Then,

for

the solution

of

^{(1.5), the a}

prioriestimate

k k

u,

(t )ll

^{2 2}

n=l n=l

k

HI

n=l

(1.10)

holds, where

tk--k7. ^<_

^{T and K is} independent

of

k<n* ande>O.

Proof

By (1.5),^{we have}

k-1

( ^{bt+l +1}

7"Z ^{(MOz/+l .{+1).}

^nt

^(LI.1

^.tA

^)H.

n=0

k-1

--(b/ff+l, b/+l)H2) TZ(f?+l,bl+l)H,.

n=0

(1.11)

For the first term on the left-hand side of(1.11), wehave

k-1

T

Z ^{(MoOoU+ 1. b/{+ )Hi}

n=0 k-1

kl²

--Z((b/

⁺¹

^--b/) b/+l)M0_

n=0

(recall

that

u ^-0).

^{Therefore, the left-hand side}

of

(1.11)

^iseasilyestimatedfrom below as

k k

H

+ llu ll /o

^/

lu2 IIg=,

n=l n=l

where

LI.1 ^_> 511.

5>0. The right-hand side of

(1.11)

is estimatedfromaboveas

k-1

TZ(f+I,bl+I)H

n=0

Tgk-1 ^k-1

S -1-

Ilfl

^n-l-1

2 _n=0

(recall the evident inequality (f,

u) _< 5/211.112 + 1/(2)[Ifll 2,

V6

> 0).

These two estimates obtained (in combination with

(1.11))

lead directly to

(1.10).

THEOREM 1.3 Let the conditions

of

Theorem 1.1 be

satisfied

^andsuppose that

(f(,

ul

)H, ^(g,

^U

)Mo, ^Vu

^{H, n}

^_<

^n*.

(1.12)

The right-hand side of

(1.14)

^{is estimated from} aboveas

k k

’T-(fln, 00U)HI "[-(g,oU)M

n=l n=l

<-

^T

Z-

_n=l

^{(IIOou{IMo + IlgllMo)}

(see (1.12)).

Thus,

(1.14)-(1.16)

^leadto Then,

for

the solution

of ^(1.9),

^{theaprioriestimate}

holds,where theconstantKisindependent

of

^k

^<_

^n*

and^6‘

>

^O.

Proof ^{By (1.9),}

^wehave

k-1

n=0

+ (Ll,U+,o,+)i4)

Tk-1

b/+ (0b/+l

6"

n=O k-1

TZ(f+I,OoU+’)H1. ^(1.14)

n=0

The terms in

(1.14)

involving

Ll,1

^and

L2,1

^{can be}

estimated frombelowinthe followingmanner:

k

n=l

n=l

By

(1.17)

andstandardinequalitiesintheoryof dif- ference methods

(see

Ashyralaev and Sobolevskii, 1994; D’yakonov, 1972; Mitchell and Griffiths, 1980; Thomee, 1997),

(1.13)

follows.

THEOaEM 1.4 Let the conditions

of

Theorem 1.3 be

satisfied.

^Then,

for

^{the solution}

of

^{(1.9), the}

aprioriestimate

k k

n=l n=l

holds, where theconstantKisindependent

of

^{k n*}

and

>

^O.

Proof

^{Inaccordancewith(1.9),we}observe that

L,,2L2,u (f Moou ^{Ll,,U). (1.19)}

By

(0.9)

and(1.6), wededuce from

(1.19)

Therefore,the left-handsideof

(1.18)

^isestimated

from above as

k

n=l

(see (1.20)).

After that, it suffices to apply the apriori estimates obtained.

Nowwe indicate apriori estimates forproblem

(1.5)

^{withe 0.}

THEOREM 1.5 Let the conditions

of

Theorem 1.1 be

satisfi’ed.

^Then,

for

the solution

of ^(1.5)

^with

e-0,theapriori^estimates

(1.21) (1.22)

hold, where theconstantKisindependent

of

^k

^<

^n*.

Proof

^{Observe that the} second equation in sys- tem

(1.5)can

be rewritten as

L2,10u

^{’+l --0.}

Hence

(1.5)

implies that

(1.23)

(see (1.14)).

After that the same reasoning as the Proof of Theorem 1.3 leads from

(1.23)

to

(1.21).

To prove

(1.22),

it suffices to rewrite the first equationinsystem

(1.5)

as

L1,2u+l f+l ^{MooU,+l Ll,lu,+l}

andapplyevidentinequality

Uff

⁺¹

< IIf

ⁿ⁺

II/-/, + IIoMo"f

^’+’

IIH, H

/

IIL, (1.24)

(see (1.20))

in combination with

(0.9)

and

(1.21).

1.3.

A

Priori Estimatesand Correctness forMore General Problems

We return now to problem

(1.8)

ⁱⁿ general setting (without^therequirement that all

f2

⁰

^(see

^prob-

lem

(1.5))"

OoMou + Ll,lu + L1,2u

2

--fl n,

L2,1 u ^{gu f2} .

Itwill also be useful to rewrite it in terms of

k k

U1

^7-

Z

_n=O

^{u, U2}

^{k r}_n=0

^{u, (1.27)}

k k

F1

^r

f, F2

^{k r}

Zf ^(1.28)

n=0 n=0

(with _ui

-f,.o

^{O, i-1,2)as}

Mou

^q-

^{LI,1U + L1,2U F,}

L2,1U CU ^F,

(1.29)

n-1,..., k

<_

n*-

T/r.

^{Itis evidentthat}

(1.26)

and

(1.30)

^are equivalent. Observe also that

Mou

$0M0 7{’,

’ ^{0 7{’, u’ 0}

k

Ull 2H ^<

^Tr

Z ^{IHinl 2H}

^i,

n=0 k

lull

^2Mo

<Zrll

^{/2i M0"}

n:0

i- 1,2,

(1.31)

THEOREM 1.6 Let the conditions

of

^{Theorem 1.3}

be

satisfied.

^Then,

for

the solution

of ^(1.25)

^and

(1.26), theaprioriestimates 82

k

/ r

(I &u[’

²_Mo ^-4-

lug’ IN

^{2 "Jr-}

^e[lU2

^2g

2) ^<

^KF,

n=l

(1.32)

k

"r

Ilu 2142 ^{<_ KFk (1.33)}

n=O

hold, where

k k

IOof I1.

Mo

n--1 n=l

(1.34)

theconstantK^isindependent

of

^k

^<_

^{n* andc}

^>_

^O.

Proof

Restriction

(1.26)

implies that

L2,1Oou-

Oof:.

^Hence,

(c:,Oou,u:).: (Oou:,:)n: (Oof[,:)n:.

(.35)

Combining(1.25)and

(1.35),

we seethat

k

x -

^n=l4-

e/2l}u2kll ^(llS0uUII

²_H2

^{20 + (L,0 uU, OouU).}

k

n=l

-(- 0o2 ^", u ⁺ ).)-r.

(1.36)

The left-hand side of

(1.36)

^can be easily esti- mated as

2

HI"

n=l

(.sv)

The first term on the right-hand side of

(1.36)

presents ^no problems since we can make use of inequality

(g 80U,)Mo ^< 1/4 II00uUII

²Mo

+ ^IIg IIMo"

²

^(1.38)

The secondterm on theright-handside of

(1.36)

^is

k

Z-- T

Z ^(00f2

^n,U

^2)H2

n=l k

Oof2 IIg Ilu2

n=l

where

[]ulIH

is estimated from above via

(1.24)

and

(0.9).

Thus,

k

(

²

(2)

²

n=l

+ ilfl]2 ^{., + IluU I.,}

²

) ^(1.39)

with a t

>

0 andsmallenougha

>

0. Combination of

(1.38)

and

(1.39)

yields thedesired estimateforY in

(1.36)

and a basic consequence of

(1.36)

and

(1.37),

whichleads to

(1.32)

and

(1.34).

Inequality

(1.33)

followsfrom

(1.25)

and

(1.32) (see

theProof of

(1.22)).

THEOREM 1.7 Let the conditions

of

^{Theorem 1.3}

be

satisfied.

^Then,

for

the solution

of ^(1.29)

^and

(1.26), theaprioriestimates k

7-

ZII llunl12

¹¹¹_Mo /

gll.,

^{2 /}

}1 gffll

²_H2

<

KFk,

(1.40)

n=l

k

7-

Z ^{f 12H2 _< K}

^k

^(1.41)

hold,where

k k

p r(i fill2

^n,

^{+ IIgTII}

²Mo

⁺ IIf2

n=l n=l

(1.42)

theconstantKisindependent

of

^k

^<_

^{n* ande}

^>_

^O.

Proof

^{This is just a repeat of} the Proof of Theorem 1.6, with obvious modifications con- nected with the use of

(1.27)-(1.31)

^{to obtain}

(1.40)-(1.42)

from

(1.32)-(1.34).

Note that all results obtained can beeasily for- mulated forother Hilbertspaces

H1

^and

H2.

^What

is of importance is that

M0

^E

(H1;H),

^M-- M*>⁰ and that Theorem 1.1 applies; hence elas- ticity and hydrodynamicsproblemswith

L2,1-

^div

can be mentioned as examples

(see

D’yakonov, 1996a and references therein). We ^{note also that} the case

M0 I1

^is allowed. Instead of I2, we can

dealwith

L2,2

^G

Z2(H2;H2), L2,2 L,

2

_>

0;

(1.43)

then

C[]U2][

²_H2in our estimatesshouldbereplaced by

llu2112

L2,2

c(L2,2u2, U2)H2.

^{Forexample,the choice}

HI al,1 (; r; r0);

^H2 H

(r; 1-’0)

2 2

IlURIIg, l"210,r0 ^(1.44)

isof^aspecial importancewhenwewish topreserve the homogeneous Dirichlet conditions on

E0

^c

^E;

here

0- f’0

^{is a union ofseveral arcs;}

HI([’;

f’0) is asubspaceof

HI(I )

whoseelements vanish on

0

and

GI,I(;I;f’0)

corresponds to a subspace of

Gl,l(f;I )

^with elements having traces in

H2 ^(see

D’yakonov,

1997b).

Notefinallythat^moregeneral problemswithnonsymmetric operatorsarepossible

(see (0.12)).

1.4. NonstationaryBoundary Conditions

Instead of the conditions

L2,1u- gL2,2u--f

ⁱⁿ

(1.5)

and(1.26),^newconditions,

L2,100u OoL2,2u

2

f2, (1.45)

areconsideredhere

(see (1.43)

and

(1.44))

^with

u:

0 for simplicity ofexposition. (The original para- bolicproblemsinstrengthened Sobolev spaces^with nonstationary boundary conditions oftype

(1.45)

mightbeof special importanceincompetitionwith thehomogeneousDirichlet conditions on

E0.)

THEOREM 1.8 Let the conditions

of

Theorem 1.3 be

satisfied.

^Then,

for

the solution

of ^(1.25)

^and

(1.45), theaprioriestimates

Ilu(ll

²

(1.46)

-

ⁿ⁼⁰k

^{Ilu ll}

^2H2

^{< KFk (1.47)}

hold,where

k

n=l

theconstantKisindependent

of

^k

^<_

^{n* and}

^>_

^O.

Proof

^{It is easy to see that the} corresponding analogofTheorem 1.1 holds. Thusthe solution of (1.25) ^and

(1.45)

^exists and is unique. To _prove

(1.46)

and

(1.47)

^we apply the Proof ofTheorem 1.6. Observethat

(1.35)

should bereplaced by

(L2,1ou, U)H2 (oU, U)L2,2 ^(fff,

no

Oo.f

^{is needed here. This} enables us to apply the Proofof Theorem 1.6 withslightalterations.

Notealsothat thecaseof conditions

L2,1)ou cL2,2u ff ^(1.48)

isanalogousto the considered one.

THEOREM 1.9 Let the conditions

of

Theorem 1.3 be

satisfied.

^Then,

for

the solution

of ^(1.25)

^and

(1.48), theaprioriestimate

k k

II0ull

²,o

+ Ilull

H1

⁺ Ilul

n=l n=l

k

n=0

holds, where

k

., ^/lg7

Mo ^/

f;IH),

n=l

theconstantKisindependent

of

^k

^<_

^n*and

^>

^O.

Proof

^{This is just a repeat} of the Proofs of Theorem 1.8 and 1.6, with obvious modifications connected with the use of

(L2,1OoU, U)H2 ^(u, U)/2,2 ^(.f2 , _U)H2

insteadof

(1.35).

2. PERTURBATION THEOREMS

2.1. Perturbation Theorems for ParabolicProblems

Here,we studydependence of the solution oftype

(1.9)

and

(1.8) (or (1.29)

^and

(1.26))

onthe param- eter c when c-++0. We denote by

z,

^{i-1,2 the}

n _u where index e is nowused to difference

uc,

0,i,

indicate the corresponding problem and its solu- tion.We alsomakeuseof

k k

n=0 n:0

THEOREM 2.1 Let the conditions

of

Theorem 1.7 be

satisfied

and suppose that all

Fk ^<_

^{K* with K*}

independent

of

^{k<_n*. Then,}

for

^{the solution}

of (1.25)

and(1.26),^wehave

k k

Mo

+ IlZl[2H1

^-4-r

IIZ ’ll = ^K2,

n=l n=0

(2.2)

where the constant Kis independent

of

^k

^<_

^{n* and}

c>O.

Proof

^In accordance with

(1.25)

and (1.26), we have

OoMoz + Ll,Z + L1,2z

2n 0,

L2,1 z ^{guen,2 f.}

(2.3) (2.4)

These relations

(2.3)

and

(2.4),

togetherwith

(2.1),

imply that

OoMoZ ^{+ L,Z + L1,2Z}

2n 0,

(2.5)

k

L2,1Z

^cy

Z ^uen,

²

^{F. (2.6)}

n=0

Hence,

Theorem 1.7 for

(2.5)

and

(2.6)

leads di- rectlyto

(2.2).

It is worth noting that Theorem 2.1 implies asymptotically optimal (O(e))convergence ^{in the}

norm the square of which is given by the left- handterm of

(2.2);

the useof Theorem 1.6 and the norm defined by

(1.32)

and

(1.33)

leads only to

O(e/r)-convergence. There is also a possibility to establish

O(el/2)-convergence

forproblem

(1.9).

THEOREM 2.2 Let the conditions

of

^{Theorem 1.3}

be

satis’ed

^andsuppose that

k

,_,,

^/

IIg7112 o) ^< (2.7)

n:-I

with K* independent

of

^k

^<_

^{n*. Then,}

for

^{the solu-}

tion

of ^(1.9),

^wehave

k

iIzll

^{2 2}

^<

^Ke,

^(2.8)

where the constant K is independent

of

^k

^<_

^{n* and}

c>0.

Proof ^{By (1.9)}

^and

^(1.5)

^{withe-0, we have}

OoMoz + L z ^{+-L 2L2 lZ{} L1,2Uo,

2.

Therefore,

2 2

Since

(u

0,2,

L2,1ZI )H, ^< -lllL2,zll,

^/

4llzTil.,

we can conclude from

(2.9)

and

(2.7)

that

(2.8)

holds.

Analogously, perturbation theorems canbe ob- tained on thebasis ofTheorem 1.8

(1.9)

for para- bolicproblemsinstrengthened Sobolev spaceswith nonstationary boundaryconditions.

2.2. Splitting ^{of the} Regionfor Parabolic Problems

We startby consideringapartition of

h

^{into aset}

of blocks (panels)

(l,..., hi,;

each domain

fi

^has

a Lipschitz piecewise smooth boundary F

i-0fi

andtheSobolev space

HI(Fi)

^canbe easilydefined (all its elements are equivalent to continuous functions).

On the set

s=[,.JiOf; (the

^{union of the panel} boundaries)^{wedefinetheHilbert}space

H2 ^H2(S)

withthe square of thenorm

Finally, we note that hyperbolic problems de- serve a separate study, but it is fairly clear that modifications of the results indicated above are possible.

i*

IIg

²

H2--Z

²

(2.10)

i=1

H2

consists of functions gE

L2(S)

suchthattheyare equivalent to continuous functions on Sand their restrictions toeach

P;

belongto HI(F;).

The modelstrengthenedSobolev space

GI,1 Gl,1 (f; S) H1 (2.11)

consistsoffunctions in

H(f)

suchthat their traces oneach

F;

belongto

H(P;),

so wemaydefine

i*

2/_/1- ]

²H1

()+ Z ^]lTrr; vII2HI(I/)" ^(2.12)

i=1

Itisknown that

H1

^{isaHilbertspace andthe trace}

operator

Trs

^E

(HI;L2(S))

^can be considered as an element of

(HI;H2(S));

^moreover,

Trs /2(H1;H2(S))

^{is normally invertible} operator

(see

D’yakonov, 1996a;

1997b).

^This key fact enables usto apply results indicated aboveto problems in the strengthened Sobolev space

H1 (see (2.11)

and

(2.12))

and in the Hilbert space H

H

^x

H2 ^(see (2.10)).

^Forexample,Theorem1.9appliesfor

(1.25)

with conditions

(on

S)

L2,100u cuff

^0;

the case e=O corresponds to split (with ^respect to the original partition of

()

problems dealing witheach panel separately (with thehomogeneous Dirichlet conditionsoneachP;);the corresponding perturbationestimate is

k k

TZ ^II0ZII2

^Mo-j-

^IlzklllH,

²

^-+-

^7-

Z

^{"2n 282}

^{< K2.}

n=l n=0

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